t 


A GENERAL  CYCLIDE 

WITH  SPECIAL  REFERENCE  TO 

THE  QUINTIC  CYCLIDE 


By 

HARVEY  PIERSON  PETTIT 

A.B.  Kalamazoo  College  1914 
A.M.  University  of  Kentucky  1919 


THESIS 

SUBMITTED  IN  PARTIAL  FULFILLMENT  OF  THE  REQUIREMENTS 
FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY  IN  MATHEMATICS 
IN  THE  GRADUATE  SCHOOL  OF  THE  UNIVERSITY 
OF  ILLINOIS,  1922 


URBANA,  ILLINOIS 


UNIVERSITY  OF  ILLINOIS 

THE  GRADUATE  SCHOOL 


_191 


i hereby  recommend  that  the  thesis  prepared  under  my 

SUPERVISION  BY. '-ettit 

ENTITLED  A GENERAL  TO LITE!  with  special  reference  to 

TIIS  QU I NT  1 0 CYCm?-. 

BE  ACCEPTED  AS  FULFILLING  THIS  PART  OF  THE  REQUIREMENTS  POR 


Recommendation  concurred  in* 


Required  for  doctor’s  degree  but  not  for  master’s 


aiOMIJ  II  HO  YriSHHVtUU 
. ■ ■ " w . 


(J  ■ 5 


, ■-]  • . VI Vj  ']?, 

■ I 


' • 


TABLE  OF  CONTENTS. 

I.  Introduction 

II.  Tbs  Quintic  Cyclide 

1.  Tbs  Double  Tangent  Cone 
a.  Direct  Development 

o.  Mapping  on  a Ruled  Cubic 

2.  Systems  of  Curves  on  the  Quintic  Cyclide 

a.  Curves  of  order  6,8. 

b.  Other  Systems  of  Curves. 

3.  Minimal  Lines  on  the  Quintic  Cyclide. 

4 . Circles  on  the  Quintic. 

5.  Special  Quintic  Cyc Tides, 
a . First  Special  D a s e . 

b , Inverse  o f Ruled  0 u o i o . 

c , Quin  t i c fi i t.  h 7 r t e x i r.  i. . . . 7 a.i  ical  Flans. 

III.  General  Cyclide 

1.  Projective  Description 

2.  Systems  of  Curves  on  the  General  Cyclide 

3.  Locus  of  the  Centers  of  the  Generating  Circles 

4.  Minimal  Lines  on  the  General  Cyclide. 

5.  Further  Generalized  Cyclides. 


15 

18 

20 

2 a 


pp 


37 


' 


■ 


I.  Introduction. 

The  subject  of  the  c ye  1 ides  has  been  more  or  Is 
ly  trefted  bv  nearly  all  writers  on  the  thee 
has  been  treated  from  various  standpoints  ar 
these  surfaces  have  been  fully  discussed.  The  particular 
surfaces  which  concern  this  pa  per  have  not, feowever, received  any 
an:  cunt  of  attention. 


or 

less  e x h a 

u s t i v e 

of 

surfaces . 

It 

t he 

p r o p e r t i e 

s of 

In  his  c c u r a : 


ur faces  m tne  year 


.i  e c l u r e s 


rr.  .~r  s~.  — r\  ca 

• 1 . It  I ' ‘ . 


n c ^ »•  "if 

- iiSvi.  ,y 


A i as  o r a i 


:•  la  ->  „ , r ret  0 3 S 0 r 1 ICC  h CO  1 D f,  3 1 0 U I-  Z h 3 

projective  description  of  the  eyclides, indicating  the  extension 
to  tbs  coin tie  eyelids, of  which  tbs  ordinary  quartic  eyelids 

/ j \ 

forms  a special  esse.'--' 

f fee  quin tic  eyelids, generated  by  a pencil  of  spheres  and  a 
protectively  related  set  of  tangent  planes  to  a cone, forms  a 
special  case  of  a type  of  quintics  discussed  by  ClsDsch^2'. 

St urm* 3 ^ , and  Noether  ^ .also  considered  the  same  type  of  ouintic 
although  they  offered  little  except  a recapitulation  of  the 
results  announced  by  Olebsch. 

In  the  present  paper  it  is  proposed  to  consider  t he 
properties  of  the  ouintic  eyelids, its  tangent  cones, some 
important  curves  lying  on  the  surface, and  some  special  cases  of 
the  ouintic.  Then  the  discussion  is  to  be  carried  over, in  some 
part, to  a mere  general  eyelids. 

Cl)  Tohoku  Mathematical  Journal.  7ol.  19,  Nos.  1,2.:! ay  1 0 a 1 
( 2 ) Die  Abb  ild  un  <’.  einer  C 1 a s s c von  FlScher  5 0 . , A b h a n d 1 u n r der 

KBni.cf  lichen  Oesellschaft  der  riissenschaften  a v.  06  ttin^en,  V'ol.15. 
( S ) Wathemat ische  ft  ns  a 1 o r V c 1 . 4 • 

( 4 ) Hatheiatische  An  r,  alen  V o 1 . 3 . 


Digitized  by  the  Internet  Archive 

in  2016 


https://archive.org/details/generalcyclidewiOOpett 


2 


IT.  The  Quintic  jyclide. 

§ 1 „ The  Double  I a n g s n t Cone. 

The  quintic  eyelids  is  represented  by  the  equation 
( i X »,D2  -i-  opn  j.  v r 2 - n 

where  a, 8, y are  linear  expressions  in  x.y.z  and  F and  G are 


"if 


X n = n 

A C.  V> 


the  form  R2-A  and  R2-8  respectively,  where  R2  = x2  + ,y2  + z2  and  A,  8 
are  linear  in  x, y,z.  The-  surface  is  pre.j eetivs iy  described  by  the 
cone  and  pe no i 1 of  spheres 
(2) 

1 he  eq nation  of  the  cone  proper  is 
( 8 ) ?.  2 - 4 e y = C 

and  the  vertex  is  the  point  a=8=y=0  which  lies  on  the  surface 
a s is  evident  from  equation  ( 1 ) , 

Any  line  through  the  vertex  of  the  cone  meets  the  quintic 
in  four  other  points  which  lie  on  the  generating  circles  in  the 
two  planes  tangent  to  the  cone  and  intersecting  in  the  line.  As 
the  two  planes  approach  coincidence  the  circles  corns  together 
sc  that  a generator  of  the  cone  meets  the  quintic  in  two 
double  points. 


Any  two  such  planes  are 
a A 2 + 8 A +■  v = 0 
a u 2 + 8 u + y = 0 


(4) 


From 

these 

two 

equations  we  have 

(5 ) 

; ; 

y = 1 : - ( A + u ) : A u 

1 — 1 

hese  values 

are  substituted  in 

squ  a 

t ion  ( 1 ) the  r e s u 1 1 i s 

(6) 

F 2 

- A PQ  - |i?Q  + AnS2 

= (F 

- AG. ) (P  - uO)  = 0 

which  shows  that  the  four  points  lie  by  pairs  on  the  generating 
circles  corresponding  to  the  values  A , u of  the  parameter. 


Then  if  the  two  generating  planes  coincide, i .e . if  A=u 
equation  (c)  reduces  to 

(?)  (P  - AO)2  = 0 (i) 
hence  we  have  the 

Theorem  1.  The  cone  3 2 - 4ocy  = 0 is  a double  tan 
Theorem  2.  Every  plane  of  the  Generating  cone  cuts  the 
quint ic  eyelids  in  a circle  and  a cubic  which  lies  on  a cubic 
eye lide. 

Consider  the  plans 

( 6 ) a A 2 + 3 A + y = 0 
w i t h the  s u r f a c e 

( 9 ) a P 2 + 8 F Q + y Q 2 — 0 

Eliminating  y between  (5)  and  (9)  we  have 


(10)  o:P? 


p ppi  ^ \ JA2  p \ p 2 ■—  ( 


A Q ) ( oc  P + a A Q 


The  second  factor  in  (1C)  is  of  the  form 
(11  ) a 'F  + D ’0  - 0 

which  is  the  equation • of  a cubic  cyclide. 

Theorem  3.  The  curve  of  contact  of  the  double  tangent  cons 
with  the  quint ic  cyclide  is  of  order  5.  v2^ 


The  equation  of 

the  quirt ic  eyelids 

is 

(12)  o-.F2  + 

8PQ  + yQ 2 = 0 

while  the  equation  c 

f t h e double  tangent 

cone  is 

(13)  32-4 

OC  Y — 0 

Multiply  (22)  by  4a 

and  rearrange  terms. 

T he  res u 1 

(14)  ( 2 a F + 

BO)2  - (3  2 - 4 o:  y ) C 2 

ii 

o 

( 1 ) The  analytic  treatment  of  theorem  1 follows  the  w c r If  of 

Clebsch.  loc.oit.  p p 9,10. 

Theorem  3 due  to  Clebsch.  The  proof,  however,  is  new. 

v ~ / 


• re. 


4 


The  curve  cf  contact  is  the  intersection  of  tne  double  tangent 
cone  (13)  with  the  cubic  eyelids 
(15)  p ocP  + 30.  = C 

This  intersection  is  of  order  6,  but  the  line  ot=B=0  factors  out, 
leaving  a proper  intersection  of  order  5. 

Theorem  4.  The  proper  single  ton§eni  cone  with  vertex 
a=?=C  is  of  order  6.  ' 1 


The- 

intersection  cf 

the  fi 

rst  oolar  of  the 

oo  i 

nt  a.=?  = y-C 

ith  the 

quin  tic  c.yclide 

is  of 

order  30, reduced 

b.y 

3,  since  the 

point  lies  on  the  puintic.  Then  the  cone  with  this  point  as 
vertex  and  tangent  to  the  quint  ids  has  this  same  curve  as  curve 
of  contact.  But  there  breaks  off  from  this  cone, the  cone  through 
the  nodal  curve , counted  twice  and  the  double  tangent  cone 


counted 

twice.  This  reduces 

t he 

^ V U ^ <-> 

y 2 s 

I '•■> 

•I- 

tO 

CO 

II 

2 and  leaves 

v»  mrs  ir»  v* 

C i.  v y v : 

s i.  n g 1 s t a n g e n t c c n e 

c £ 

order 

6 . A 

direct 

analytic  proof 

of  this 

theorem  i s given  by 

Gleb 

s c h . 

Theorem  5.  The  c.urve  of  contact  of  the  single  i anient  cone 
with  the  Quiniio  cuclide  is  of  order  7.  (2) 

The  defining  curve  of  the  single  tangent  cone  is  of  order 
6.  Then  any  plane  through  the  vertex  meets  this  curve  in  six 
points.  But  since  one  of  the  generating  circles  passes  through 
the  vertex  one  line  through  the  vertex  is  tangent  to  the 
surface  at  that  point.  Bence  the  vertex  itself  must  be  counted, 
as  one  point  in  the  intersection  with  the  curve  of  contact. 

Thus  the  curve  of.  contact  is  of  order  7. 

(1)  Clcbsch  loc.cit.p  10. 

( 2 ) ibid. 


. 


Theorem  6.  The  locus  of  the  centers  of  the  generating  cir 
circles  is  a space  curve  of  order  5. 

In  order  to  determine  this  locus, consider  a line  through 
the  center  of  one  of  the  generating  spheres  and  perpendicular 
to  the  corresponding  plane.  The  intersection  of  this  peroendicul 
lar  with  the  corresponding  plane  of  the  tangent  cone  is  the 
center  of  the  circular  generator. 

The  equation  of  such  a sphere  may  be  written 
(16)  R 2 ( 1-A  ) -2w -j x-2\y ? y-2>i/ R z-y/4  = 0 
w here  v % , w P , v/ a , v/ 4 are  de f ined  b y 

V 1 h ) 4 f z ~ 1 A - 1 , 2 p P ~ >1  ? A 3 p , 3 W yj  — a 3 A -•  R , '•}/'  4.  — H 4 A 4 


rhe  suoscra 


1, 2,3,f  indicating  respectively  the  x,y,z  and 


constant  coefficients . 

The  center  of  this  sphere  is  the  point 

?s;  . Tfr  ^ )!•  _ 


(20) 


-A'  1-A 


The  cor responding  plane  is 


(21) 


r A2  + ?X+ y = ® a.x  + ©2  y+a>3z  + <p4  = 0 


using  the  same  subscript  notation  as  above . It  is  evident  that 


the  cd's  are  Quadratic  in  A and  th; 


The  direction  cosines  c: 


1 s linear  in  A . 


-lane  (21)  are  proportional 


the  coefficients  tp*, Then  the  equation  of  the  line  in 


iuesticn  is 


( 1 -A  ) x— y> , _ ( 1-A  ) y-u ; o _ ( 1-A  ) z—\y 


(22)  

Solving  scuations  (21)  and  (22)  simultaneously  for  x,y,z 


we  find 


cn  cn 


cx  = 

( 1— A ) cp  x cp4 

Cp  1 ( 'P  ? W c> 

* 

) 

~ Ti 

(<pf 

+ cp? ) 

( 9 ' 

) = 

(1-A  )cppcp4 

+ <P  ? ( cp  ! v{/  , 

T 

PpWp 

) 

- <P? 

( m 2 

\ y 3 

+ cp? ) 

V 2 C . 

DZ  = 

(1-A  )fps<p4 

+ 9 ? ( CP 1 w n 

Cpp'4'2 

\ 

j 

~ <Ps 

(cp? 

+ ®|) 

P t = 

(l-A)(cp?  + 

qp  | + <p  | ) 

and 

since  ti 

1 e co 1 s are 

quadratic 

a 

nd  th 

& 

•j; 1 s 

lin 

e a r i n 

expresses  parametrically  the  equation  of  a space  quin tic . 

Theorem  7.  The  normals  from  the  centers  of  the  spheres  of 
the  generating  vend l to  the  correspond  inf  fenera tint  planes 
form  a ruled  cubic. 

Consider  any  plane 
(24)  ax  + by  + 2 + d = C 


and  the  line 


( 3 5 ) 


( 1-A ) cppX  - ( 1 — A ) ® a y 
( 1-A ) m 3 x - ( 1-A ) c x z 


+ X 2 w , 

+ D - \y . 


Cl'.  „ w , = 0 


(26) 


Eliminating  z we  have 

( 1 — A ) 0/  2 x - (1-A)q;iy  + y x y 2 - a ? w . = C 
(l-A)<j)sx  f (l-A)q>.1(ax+by+d)  + <pi«|r3  - sj3 w - = 
From  which 

( 2 7 ) px  = b ( cp  ? \y  4 - cp  A y s)  *.  © , \ij  5 - m a p p - ( 1 - A ) d cr , 


0 y = a 


v ’.p  p p P I J • Pp 


"’p  - (l-A)dp. 


T bus  any  pi; 


( a vii  - D cp 

P Vf 

) 

r -j  Q n r~  -p 

t he 

ruled 

s ur f ace  formed 

by 

t hese 

r + h r»  a g 

and 

‘p  op20 

the  ruled  surf a 

0 e 

itself  is 

a cubic 


The  plane  (24)  cuts  the  places  of  the  double  tangent  cone 
in  a class  conic  of  which  there  are  two  lines  through  every  point 
of  the  cubic  cut  cut  by  the  ruled  surface  of  Theorem  7.  On  the 
other  hand, any  line  of  the  class  conic  cuts  the  cuoic  in  three 
qoints.  Hence  there  is  a three-two  correspondence  between  the 
ooints  of  the  cubic  and  the  lines  of  the  conic.  Then  there  are 


2.  Systems  of  Curves  on  the  Ouintic  Cyclide. 
a.  Curves  of  order  6,9. 

theorem  8 . There  are  -1 7 a u int ic  eye l ides  in  space. 

A quin tic  cyclide  is  determined  by  the  pencil  of  spheres, 
the  double  tangent  cone  and  a pr objectivity  between  the  spheres 
of  the  pencil  and  the  tangent  planes  of  the  cons. 

The  radical  plane  of  the  pencil  of  spheres  can  be  chosen 
in  *s  ways.  Having  chosen  the  radical  plane, one  can  choose  the 
radical  circle  in  ~3  ways  and  then  the  pencil  of  spheres  is 
completely  detsrmi ned . 

The  vertex  of  the  double  tangent  cone  can  oe  chosen  in 
303  ways  and  the  base  of  the  cohe  in  «5  ways. 

Consider  any  three  planes  of  the  double  tangent  cone. 

Each  of  these  is  cut  by  the  spheres  of  the  pencil  in  -1  circles. 
T he  c ho ice  of  any  one  c i rcis  in  each  of  t he  t hree  planes 
fixes  the  project  i v i t y w h i c h vr.  a y t here  f ore  be  set  up  in  *s  w a y s . 
Therefore  t h e .c  ui  n tic  c v c 1 i d e m a y o e d e t ermine  d i n 

cc  3 • cc  3 • oc  3 • ccS-cc3  = cclV  n 3 V S • 

Theorem  9.  Any  sphere  cuts  the  quint io  cyclide  in  the 
sphero-circle  at  infinity  and  a pi. rite  sphero-sextic  which 
lies  of  a.  ruled  c u b i c . 

The  equation  of  the  quint ic  cyclide  may  be  written  in  the 

form 

(1)  q:,r  - a(R2-::)2  + 8 (? *- V)  (3 2-S ) + v ( d 2 -8  ) 2 = C 
and  any  sphere  in  the  form 
( 2 ) R 2 — C = 0 

Eliminating  Hz  between  the  two  equations  we  have 


. 


b . Mapping  on  a Ruled  Cubic. 

Consider  the  mapping  of  the  quintie  eyelid; 

(1)  aF2  + 5PQ  + yC2  = 0 
by  means  of  the  transformation 

(2)  x'  = ^ 

Y 


y 


y 


f - 


Then  the  surface  ( I ) goes  ever  into  the  surface 

( 3 ) X ’ z ’ 2 + y ' z ' + 1 = C 

a ruled  cubic  whose  era  tors  are  parallel  to  the  x 1 y ’—plane 
and  tangent  to  the  parabolic  c y 1 i n d e r 

(4)  y ' 2 = 4x ’ 


since  t 

h e y p r o j e c.  t c n t h a 

1 y '—plan 

to 

the 

1 i n ° ^ 

nvsl  o 

ping 

the  par 

a b o 1 a i n w h i e h t h s 

c 

y 1 i n d e r 

(i) 

c u t 

s t h 

at  p 1 a n 

e . 

T b 

e transformation 

r o 

m t h e I 1 

-spa 

to  t 

re  ~-so 

a c e i 

s a 

1 -2  transformation.  1 r a t 

I s 

f q •:.}  cj  c 

ngls 

'0  0 

int 

0 L 0 ll  v 

ruled 

cubic 

there  c 

or respond  two  poin 

is 

of  the 

q u i n 

tic 

-j  ' o 
a _J  or 

c h l5  -s  n s 

r a t o r 

of 

the  cyl 

i n d e r ( 4 ) is  • 

nt 

to  the 

0 U b 1 

c , 

Then 

~ p 0/ 

enera 

tor 

of  the 

cone 

(5  i 

22  - 4 a.  y - C 

w h i c h c 

orresponds  to  (4) 

is 

d o u b 1 y 

tang 

ent 

t o 

the  q ui 

d t i c . 

A nd 

each  circular  generator  of  the  quintie, which  corresponds  to  a 
generator  of  the  ruled  cubic, is  tangent  to  the  cons  (5)  ir  two 

points  . 

Other  properties  of  the  quintie  cyclide  could  be  deduced  f r 
from  this  mapping.  It  has  seemed  more  advisable, however,  to 
proceed  by  direct  methods. 


( ■< ) 


■ , - 


(C— A ) (0-8  ) t(  J-5  - 


which  is  a ruled  cubic  genera tea  by 


(4) 


V 


_ ‘ ■ _ , 


V ~ H ) 

and  which  has  for  a double  line 

(5)  C - k = 0 - B = 0 

The  ore  or:  20 . Any  sphere  cuts  the  quint  ic  eyelids  in  a so  hero 
sex  tic  having  two  double  points  and  which  can  be  projected,  into 
a plane  sextic  of  fenus  2. 

By  checrea  7 any  sphere  cuts  the  quin  tic  eyelids  in  a 
spbero-sext ic  which  lies  on  a ruled  cubic.  But  since  the  ruled 
cubic  has  a double  lire  ns  curve  of  intersection  has  two 
d o u d 1 s noi  r t.  ? r . a pa  r r - ■ o ■ • -•  ■ p - ;r  - •-»  i p i.  ? o r r>  = > ■ r r-  r • - p 

s 7 3 r V D 0 1 T: ' 


i 


the  ruled  cubic  which  necessarily  t 


en era tor s 


s sphere  in  00111x0 


Xr*  ! 

il  v 


urve  of  intersection,  'then  if  the  curve  09  pro.] 


on 


any  plane  from  a point  of  the  double  line  of  the  ruled  cubic  tbs 
result  is  a plane  sextic  with  a multiple  point  of  order  4 and 
two  ordinary  double  points.  But  the  multiple  point  of  order  4 
counts  for  6 double  points  and  then  the  sextic  has  8 double 
ooints  and  hence  is  o.  genus  2 . 

The  double  line  of  the  ruled  cubic  lies  in  the  radical 
plane  of  the  pencil  of  generating  spheres  and  hence  the  double 
points  of  the  spbsro-sextic  lie  on  the  nodal  circle  of  the  ouint 


ic  . 


Equation  (1)  can  be  rearranged  in  the  form 
( 6 ) co  * = (a+ 3 + y ) 3 4 — ( 2 a A + 6 ( A + 3 ) + 2 y B ) R 2 + a A 2 + 3 .A  B + v 3 2 = 


11 


and  we  then  have  the  corresponding  form  for  the  ruled  cubic 

(7)  Rs  = ( a+o  + y )C2-.[ 2aA  + B(A  + 3)  + ?y3]  C + a.4 2 + f3AB  + yB2  = 0 
Subtracting  (?)  from  (6) 

(8)  <ps-Rs  = ( a + 6 + y ) ( R4~C2 ) - [ 2<xA  + 8 ( A+8 ) +2 y3] (R2-0 ) 

= ( 1 $ -0  ) [ ( a + ? + y ) ( d 2 +C )-[  .2  a A + ?•  ( A + 3 ) + 2 y 3 ] ] 

= (R2-a)f>, 

where  is  a cubic  c.vclide 

(9)  a (R 2— 3 A+C ) + ? ( 3 2—a— 3+0 ) + y ( 3 2— 38+0 ) = 0 
generated  o y 

, , (a*K)  + A(|  + v)  = C 

(10) 


(R2  - 


^ A 


) _ A (R2  - 23  + 0)  = 0 


and  which  cuts  the  quintic  eyelids  in  the  line 

(11)  A " 3 = ° 

<x  + 3 + y = 0 

From  equation  (8)  we  have 

(12)  err,  - R ~ + (R 2 - 0)(pa 

That  is, the  quintic  eyelids  may  be  generated  by  the  pencil 
c u o i c s 


(13)  R ..  + A «pR  ~ 0 

and  the  pencil  of  concentric  spheres 

(14)  R 2 — 0 - A - 0 

8 u t t fe  is  ns  nc  i 1 of  co  n c e n t r i c s o h s r e s c a n be  chosen  in  ®=3  tv  ays  . 
And  since  the  choice  of  0 fixes  R,  and  we  have  the 


Theorsfli  11.  f k e 

quintic  cyclide 

may  be  generate 

d in  a Iri p l y 

i n f i n it  e n u in  bsr  o f 

ways  by  a pencil 

of  cub  ics  ana  a. 

p r o J e c i i u e l y 

related  pencil  of  c 

oncentric  spheres 

• 

Theorem  12.  The 

whole  system  of 

sphero-sext ics 

l a in g o n 

ruled  cub  ics  can  be 

obtained  as  the 

intersection  of 

the  a u i n tic 

12 


eyelid es  and  spheres . 

There  are, according  to  Theorem  8 == 17  quin tic  eye Tides  in 
space.  There  are  in  space  0=4  spheres.  Then  we  should  expect 
*=-21  sphero-sext ics  as  the  intersections  of  the  spheres  with 
the  eye  tides. 

There  are  == 1 3 ruled  eubics  in  space.  A r u 1 a d cud  i c is 
determined  by  two  lines  in  space  and  a [ 2* 1J -correspondence 
between  their  points.  The  two  lines  can  be  determined  in  °=8 
ways  and  the  [2*1] -correspondence  can  be  as  ablished  in  <*s 
w a y s , making  ocl3  w a y s c f c b c o s i n g t r s ruled  c u Die. 

Bence  there  are  actually  only  ocl7  sphere— sex  tics  of  the 
required  type. 

The  apparent  contradiction  disappears  when  it  is  seen  that 
there  are  quint  ic  eye  tides  through  any  such  sphero-sext  ic . 


lonsider  the  spher< 


tic  with  two  double  points,  r be 


lie  on  the  nodal  circle  of  the  quin  tic  c.yclide  and  on  the  nodal 
line  of  the  ruled  cubic.  The  ruled  cubic  is  unique  because  its 
double  line  is  fixed  and  all  its  generators  are  determined. 

The  sphere  is  unique  because  an  infinite  number  of  its  non— 
ccplanar  points  are  fixed. 

The  ruled  cubic  may  be  written  in  the  form 
(15)  <x  (.4-0) 2 + B'(A-O)  (B— 0)  + y(B-C)2  = 0 
With  A , B , 0 , o: , ? , y fixed,  the  ruled  cubic  and  the  cubic  eyelids 
and  hence  <pr  are  determined.  The  question  must  then  be  consider 
ed  as  to  the  number  of  ways  in  which  R,  can  be  represented  in 
the  form 


(16) 


( A 1 -0 ) 2 + 3,(A,-C)(B.,-C)  -i-  y'(B'-O) 


2 = 0 


13 


Since  the 

nodal  line  of  R 

3 is 

f i x e d 

we 

may  find  A'  ana  B!  in 

following 

w a y . 

(17) 

A ' - C = A x ( A 

- 0) 

+ u a ce 

- 

C) 

B ' - C - A p ( A 

- ^ ; 

4 u ? ( B 

- 

C) 

Substituting  in  equation 

(16) 

w e n a v 

rrv 

(18) 

or'[Aa(A-0)  + «4(3 

-C)] 

2 + 3 ' [ A 

(A- 

-Ou,(B-0)]  [X.(A-C)  + Mp 

+ y 1 [ A ? ( A-0 ) + !iP( 8-0 ) ] 2 - 0 

Collecting  in  posers  of  (A-C)  and  (3-C)  and  comparing  with  ( 
■?i  g find 


(18) 

a 1 

A ? + 3 ' A a A 2 + y ' A | = a 

2 a 

J A - [i i + 2 1 ( A,  a ? + A ? u , ) + 

a ! 

U f + 3 ' JJ  1pj>  -r  V ' f.i  | = Y 

S o 1 v i n g t h 

5S3 

3ous  t i. o n s for  a 1 , 3 ' . y 1 

a u \ — BArpj  + y A f 

( 2 J ) 

a ' 

( A , U p - A p M a ) 2 

,q  i 

-2aUaUp  + ;3  ( Aifip  + ApUi  )• 

( A a u g - A , a i ) 2 

T 

o:  u f - 8 A a u 1 + Y A ? 

( A 1 }'  o ' ' p A j ) 


which  expresses  o: ! 1 , y 1 explicitly  in  terms  of  X%, u x>  u?. 

Then  since 

(21)  A 1 = A,  (A-C)  + ip(B-C)  C 
B'  = A ? ( A -0 ) + (3-C)  + 0 

are  con-homogeneous  inA1,  A?,  ua,  tiP  there  are  4 essential  para 
and  hence  3=4  ways  of  ?ir i ting  the  s am®  ruled  cu.b ic  in  the  req 


f or  m . 

But  for 

e v s r y 

such  c 

h o i c s 

of  the  Rs  there  is  a unique 

q u i n t i 

c c yc  1 i 

de  and 

h a n c ° 

t here 

are  3=4  q uint ic  eye  1 ides  thro 

every 

sphere- 

s e x t i c 

of  the 

t y p e 

under  consideration. 

Therefore  the  number  of  apparent  sober o-sex tics  cc?1  n us 
be  reduced  to  =17,the  total  number  of  such  curves. 


t he 


(3-0)] 


1® ) 


meters 

u i r ed 

ug  h 


15 


b.  Other  Systems  of  Curves. 

Theorem  13.  It  is  possible  to  secure  infinite  systems  of 
curves  of  any  order  equal  to  or  treater  than  2 on  the  quint ic 
cycbide. 

We  have  already  seen  that  there  exists  a real  straight  line 
on  the  quintic  cyclide.  We  shall  see  in  a later  section  that 
there  exist  six  pairs  of  minimal  lines  and  one  real  pair. 

Any  generating  sphere  of  the  quintic  cuts  the  quintic  in  the 
nodal  circle  and  a circular  generator,  fence  there  exists  on  the 
cyclide  an  infinite  system  of  circles.  We  shall  also  see  that 
there  exists  a finite  number  of  circles  net  belonging  to  this 


s y s o e m . 

Any  plane  of  the  generating  p 
in  a circle  and  a cubic  which  lies 
exists  an  infinite  system  of  plane 

If  a sphere  be  passed  through 
the  remainder  of  the  intersection 
q u ar  t i c whose  degeneracy  gives  r i s 
set  of  circles  on  the  surface.  If 
line  of  the  quintic  tbs  remainder 
clans  quartic. 

In  general  a plane  intersects 


lanes  intersects  the  cyclide 
on  a cubic  cyclide.  Be  nee  t her 
cubic s on  the  quintic. 
one  of  the  generating  circles 
with  the  quintic  is  a sphere— 
e to  an  interesting  finite 
a plane  be  passed  through  a 
of  the  intersection  will  be  a 

the  quintic  in  a curve  of 


order  5, 

We  have  already  noted  an  important  class  of  s ext ics  lying 
on  the  quintic.  A second  class  of  sextics  may  be  obtained  as 

follows  . 

A cubic  cyclide  intersects  the  quintic  in  the  sphero-circle 


1 6 


and  a finite  curve  of  order  11.  If  the  cubic  be  mads  to  pass  tb 
through  the  nodal  circle  the  curve  of  intersection, proper, is 

cubic  be  made  to  pass 


reduced  to  order 

I 

< . 

I f , i n 

addition, t 

through  the  line 

h _ R 

(03) 

- 

0 

a + 9 

+ 

V = 0 

the  curve  of  inte 

r s 

action 

is  reduced 

Such  a cubic 

3 

yelide 

is 

(24)  (a  + 

o 

E 

+ Y - 1 

) d o - r\ 

} u r bn  — V./ 

Consider  its  intersection  with  the  quintic  eyelids 

(25)  o:  F 2 + 8PQ  + yQ.2  = 0 

Eliminating  P from  equations  (24)  and  (25)  we  have 

(26)  (Kp2  + eP2(l-a~o-v)  + Yp2(l-a-t-Y)2  = o 


C2f, 


at  + 4 + v — H ( a +3  + y ) + v ( r.  * 2 + y ) ' — >'  y ( a + : + y ) j =0 
P 2 ( a + 2 + y ) II-  2 y + v 2 — : ( 1 — y / — a ( 1— y ) + o:  ,1  = C 
P 2 ( a + 3 + y ) [ ( y-1 ) 2 + ( y— 1 5 ( a+ ? ) + a] =0 
From  which  we  see  that  the  complete  intersections  consists  of 


70— circle 

and 

the 

nodal  c 

ircle  counted 

t w 1 ' 

and  the  inter 

see 

t i o n 

of  the 

cubic  with  a 

r uled 

The  existence 

of 

s e d 

tics  b a s 

already  been 

seen. 

Octic  curves 

can 

b S 

ob  tained 

by  pass x n g a 

cub  ic 

■yclide 


through  a circle  of  the  quintic  and  the  straight  line 


A special  class  of  nonics  has  been  noted  above.  Others  can 
be  obtained  by  passing  a.  cubic  eyelids  through  a gsnsratic 

circle. 

Curves  of  order  10  can  be  obtained  by  passing  cubic  cyclid.es 
through  the  straight  line  of  the  quintic. 

Curves  of  order  11  occur  as  the  general  intersection  of  cubic 


■ 


1 '/ 

c vc 1 ides  with  tbs  quin tic  eyelids. 

It  is  apparent  from  the  preceding  that  curves  of  inter sec t ioj 
of  surfaces  with  the  quintic  eyelids, of  order  lower  than  the 
natural  intersection, can  be  obtained  by  passing  the  intersecting 
surface  t fa  r o ug  the  nodal  circle, a gen  e rating  circle  and  a 
straight  line  either  singly  or  in  combination.  This  reduces  the 
problem  of  finding  a curve  of  any  order  on  the  surface  to  the 
problem  of  expressing  its  order  n in  terms  cf  the  modulus  5. 

n = 5x  - y 

y may  be  made  to  take  the  values  1,2,3, 4.  x is  the  order  of 
the  required  intersecting  surface.  It  is  e v idsn t t h e n t h at  n 
can  always  be  expressed,  in  the  manner  required  and  hence  that 
a curve  of  any  desired  order  can  b-s  found  on  the  quintic . 


16 


3=  Minimal  Lines  on  the  O.uintic  Gyclide . 


Theorem  14.  There  are  six  pairs  of  minimal  lines  on  the 
quint  ic  eyelids . 


(1) 

(2) 

f hen  t h 


(3) 


» y*  o 


de  r 

he 

condition 

t hat 

H ^ /-> 

i i •_* 

orrs 

spending  s 

o he  r 

- A 

,o 

n 

1-A)  -2 

•!'  a x 

■A2  + 

6 A 

►h  rr  <r\  v 

i i ^ 

+ 9? 

■eq  ui 

red 

condition 

is 

—A 

r\ 

' ! • i 

Y 2 *?  I 

7 3. 

O 

1-A 

0 if. 

Ud  p 

r\ 

0 

1— A Wr ( 

\p  3 

T i 

T ? 

\lf  p «;  4 

14 

T i 

(P  p 

CP 3 Cp  4 

n 

v 

•andeni 


ne  q s are  oi  degree  2 in  A and.  the  v;  s of  degree 


7 in  A.  If  th 


O Oi  \.r 


x d a r.sio  n 


1 in  A this  equation  is  of  deg  res 
carried  out  it  is  found  that  '■ -A  : s a factor.  He  nee  A=1  is 
one  root  of  the  equation.  3ut  the  sphere  corresponding  to  A=1 
is  the  degenerate  sphere  A-8=Q  and  hence  the  pair  of  lines  in 


this  particular  case  consists  of  the  ling 
A— 3 = 0 

0!  + f.  ~ Y = O' 


(4) 


and  the  line  at  infinity  in  the  clans 
o'  + ? + y — C 

The  other  lines  of  the  curve  of  contact  are  ‘the  six  pairs  of 
minimal  lines. 

The  existence  of  the  six  pairs  of  minimal  lines  may  also 
be  established  in  another  manner . 

Find  the  coordinates  of  the  center  of  the  sphere  and  set 
uo  the  condition  that  the  radius  shall  be  equal  to  the  distance 


4.  Circles  on  the  Guintic. 

Pass  a sphere  through  any  generating  circle  of  the 


quin tic  eye  lids, e .g . through  the  circle 
P - A*  = C 
a A2  + 8 A + y = 0 


(1) 


where  A is  considered  as  fixed.  Such  a sphere  is 

(2)  P - AG  - u ( a A 2 + 8 A + y)  = Q 

This  sphere  cuts  the  quint ic  cyciide  in  the  sphero-circle  and 
a rest  curve  of  orisr  four  which  can  be  obtained  as  the  inter- 
section of  the  s p h ere  w i t h ?.  c u a. d r i c , 

Trite  the  cyclide  in  the  fern 

(3)  a (R  2-A ) 2 + 8(R2~A)(32~3)  + y(32-3)?  - C 
and  the  so  he re  in  t he  f c r m 

(4)  S - R 2 ( I— A ) - - * Ac  - u ( a A 2 + 8 A + y ) - C 
Eliminating  3 2 from  eq  nations  ( : } and  (•:')  we  have 

(5) 


a f -4  - AS*  u(a\z-r  A * y ) _j-|  2 

+ a r A — A 3*  u(  o A2e 8 A + v ) i r A~AS t uj  q; A 8 hzyl 

~1-A  1-A 

v r A— AB+ji  ( cAf_“3Ai:v  )_q  i 2 - ^ 

1-A  ~ J 


B] 


or 


( 6 ) C£  [ A ( 5-3 ) + u ( 0:  A 2 + - A + v ) ] 2 + 

+ 8 [ A ( A —8  ) + u ( 0:  A 2 + 8 A + y ) j | A-B  + u ( $ A 2 •*  3 A + y ) J 
+ y [ A -8+y ( a A 2 + 8 A + y ) ] 2 = 0 

E x p a n d i n g a n d g r o u p i h g ' t e r is  s w e 1 h a v e 

(7)  u 2 ( ot  A 2 + 8 A -I-  y ) 2 ( a + 8 + y ) + u ( a A 2 + -8  A + y ) [ 2 a A 1 2 -(•  A f 1 ) * 2 y J ( A 

+ ( A — B ) 2 ( a A 2 + 8 A + y ) = 0 


21 


This  reduces  to 

( 8 ) ( a A 2 + ? X+ y ) [ u 2 ( <xX  2 + ? X+ v ) ( a + 3 + y ) + u ( A - 8 ) ( 2 k X + ? + ? X 

-t-fl-B)2]  = 0 


(8)  ( o: X 2 + 8 X + y ) K ' = 0 

That  is, the  intersection  of  the  sphere  with  the  ouintic  cyclide 
consists  of  its  intersection  with  the  plane 
<xX2  + 3X  + y = C 


and  with  the  quadric  Kr . 


a n c 1 -c*  i 

J fi  i-  ^ -L  ^ O : 


any  quadric  through  thi 


auartic  curve  or.  intersection 


(10)  K 


v n.  = 


or  e xp 1 ie i t ly 

(11)  (.4-5) 2 + u(  A -8)  (2a  X + P+8A+2v)+  u 2 ( c: + 3 +y)(  a X 2 + - A + y ) 

— v l y 2 ( 1 — X ) — A + X b — ,u  ( a X 2 + v X + y ) ] = 0 
;oef f icisnts  of  the  various  terms  in  the  ouadrie  (11) 


7'h 


contain  X, u, v in  different  powers 

table. 


ndicated  in  the  following 


;r  ffl 


X 


(12) 


y 2 
z 2 
xy 
xz 
y z- 
x 
V 
z 

const a n t 


A > 


■2  1 


up 


A necessary  and  sufficient  condition  that  a quadric  break 
into  two  planes  is  that  every  section  z=c  shall  be  a 


degenerate 


That  is,  if  the 


22 


conic.  That  is, if  the  equation  be  arranged  in  terms 
of  x and  y the  discriminant, which  contains  z, shall  vanish 
identically. 

Let  the  equation  of  the  quadric  be, for  example 
(13)  a31x2  + 2 a 3 P x v + a. , , y 2 + 2 a1sx  z t2a1Ax  + 2 a ? , y z + 2 a 9 A y 

+a3Sz2+2a34z+a44  = 0 

This  can  be  arranged  in  the  form 


(14) 

a . i x 2 J--  a 1?x y 

( a , o.z 

"i* 

a , 4 ) x 

+ a 3 ? x y + a ? ? y 2 

~r 

( a p 2 z 

-r 

a 2 4 ) v 

+ (a,,2+a,  Jx  + 

(a 

p p z + a 

P 4 

) y + ( a p ^ z 

and  the 

d i s c r i m i n a n t i 

(15) 

3.  2 

3. 

^ p Z 1 fi  ^ 4 

ai? 

2 2 

3. 

2 3 z + a p 4 

3.ijZ  + S^4  3 j ? Z 

+ a 2 

^ * s 

s Z 

2+a~„z+a„  • 

^3  4 ^'4  4 

Setting  this  13  ' i ual  to  ’ and  : i ' th 

resulting  equation  in* power  of  z 

g J \ Si  3 3.  p O o ” ■/  1 c 3 o •;  ‘ O ^ n p ^ r;  Ci  o c ""  - • - -j  *5.  p o ‘ Cl  i «j  v.  n o c;  ■“  d.  a ^ c3.  ^ rr  / Z 

x • _ - 

c *3  C - - p - p -j-  C,  O O — ' ;1-  P P p — . P O 

r O i p <3.  p p 44  w 1?44  -1  1 ? ^ 1 4 ? 4 •'!!'•?  4 v 1 9 • 9 4 v 14  ^ 

Then  each  poyncnaial  coefficient  must  vanish  identically.  This 
gives  rise  to  three  equations  in  A , q,  v whose  degrees,  found 
by  making  use  of  the  table  (12),  are  expressed  symbolically  by 
[ A 6 q 6 v 3 J = C 
(17)  [ A 6 {i 5 v 3 } = 0 

[ A 6 q 4 v 3 ] = C 

three  equations  of  degree  15,44,13 


These 


respectively,  b are e 


MM 


24 


5.  Special  Quint ic  OycIicI.es  „ 

Theorem  16.  The  quint ic  cycoide  is  not,  in  General, 
anal  la£ma tic. 

irs  sfc  ' to  the  Special  Pentaspher ical  Coordina 
by  means  of  the  transformation^1' 

2 A x = 2«.  xt  A ( x 2 + y 2 + z 2-S  2 ) = RT6,  x, 

Ic  if  V ]r 


A (x 


+ y « + 7,  •-  + 


A2)  = -i.R2si,x 


(1)  2 Ay  = SPkxk 
2Az  = Oykxk 

the  equation  of  the  q air. tie  eyelids  becomes  of  the  form 

(2)  2ukx3  + 2uk4xfx.>  :.u1.)Ux:,Xi1xi  = C i , 4 , k=l , 2, 3, 4,  S . if/jfk 


I n o r d s r 

f o r 

the  s u r f a c s 

anallagmat ic 

it  must!  be 

p o s s i b 1 e t o f i 

7}  j ? 

n o r t h o gone 

1 transformation  of 

such  a nature 

that  there  is 

d o 1 

east  one  of 

t he  f i v e v ar i ab  1 e s 

Tf  H n r»  p» 

/V  I:  1 'w*  1J  P vt-  III  b 

only  in  even  powers, since  an  inversion  in  pentaspher ical 
coordinates  with  respect  to  a base  sphere  amounts  only  to  a 
change  in  sign  of  the  one  variable  corresponding  to  that  sphere. 
That  means  that  for  one  variable,  say  x^the  third  and  first 
powers  must  vanish.  But  this  involves  eleven  conditions.  In  the 
linear  transformation  on  five  variables  there  are  twentyfour 
essential  conditions  at  our  disposal.  Cf  these  fifteen  are 
required  to  make  the  transformation  orthogonal . (z ^ Bence  there 
are  only  nine  conditions  open  for  choice  and  there  are  eleven 
conditions  to  be  fulfilled.  Therefore  a transformation  of  this 
kind  is  not, in  general, p os sib le . 

(l)  D a r b o u x : Geometric  Analytioue  p 385 

(g)  Boeder  Introduction  to  K i g h e r Algebra  p 154  note. 


Bb 

There  are, however, a few  special  quintic  cyelides  which 
ire  anallagmatic. 


1.  Consider  the  quintic  eyelids  generated  b.y 
x a 2 + y a + z = 0 

^ n 
■>'  - 0 


(1) 


where 

( 2 ) r = x2  + y2  + z 2 — r 2 

Q.  = x 2 + y 2 + z 2 + r 2 = 0 

so  that  the  pencil  of  spheres  is  a concentric  pencil  with  the 
center  the  vertex  of  the  double  tangent  cons.  It  is  at  once 
apparent  that  the  eyelids  is  symmetric  with  respect  to  the 

origin. 


Now 

i f W 3 

i n t 

(3) 

t_ 

y 1 

Q - 

y? 

X = 

ys 

y = 

y. 

2 = 

V 5 

the  equation  of  this  quintic  eyelids  takes  the  form 

(4)  y a y I + ytJiSa  + ysy§  = o 

An  inversion  with  respect  to  one  of  the  base  spheres  amounts 
to  a change  in  sign  of  the  corresponding  variable.  Hence  the 
c.yclide  is  unchanged  by  a.  double  inversion  on  the  base  spheres 
y , y P . But  this  is  merely  the  condition  for  symmetry  with 
respect  to  tbs  origin. 

As  we  have  seen, it  is, in  general, impossible  to  find  an 
orthogonal  substitution  which  will  leave  the  equation  of  the 
quintic  eyelids  with  one  variable  occurring  only  in  even 
powers.  In  the  special  case  which  we  are  considering, a special 


transformation  can  be  found 


It  will  be  noted  that  the  equation  is  quadratic  in 


Consider 

then. 

a transf or mat ion 

(5) 

y%  = 

ay{  + by 4 

.y?  = 

cyi  + dyJ, 

ya  = 

ey4  + 

f .V  4 

+ §y.4 

.¥4  = 

by  4 + 

. j y I 

lyi  + 

m y ,! 

+ ny  4 

A p p 1 y i n g 

this  transf or mat io 

n to 

e q u a t i 0 n ( 4 

N 

/ 

w e h a v e 

(6) 

(s.y4 

+ f y \ + g y I, ) ( a 2 y 4 

2 + 

9 Q W T7  f 77  I -L  K 2 

Ci  wb  ,y  1 .y  p • o 

vl 
•'  2 

2) 

+ 

(by. 4 

+ jyj.  + ky  4 ) (acy  \ 

2 

(ad  + b c ) y i y 

7 

9 

+ b d y 4 2 ) 

+ 

(i.y4 

+ my  J.  + ny  4 ) (c  2y  4 

2 + 

2cdy  4.y  4 + ds 

y.4 

2 ) - r, 

J 

A r r 2.  n g i n g 

t his 

-5  y-»  y\  /p  >7  O V»  -r*  /“V  +*  T7 

.1.  LJ  L_,  v_/  .J  w . ; *)  7 

. > 'll 

nd  suppress! 

ng 

primes 

facility  in  w r i t in'-, ; s h 3 v g 

( 7 )  v 2 i ( a 2 s + a c n + c 2 i ! y ,,  +•  f.  n ? ? » /:  -*-£  •-  x ) y . + ( a 2g  + ack+c  2 n ) y r j 

+ y xy2  [ (2abe+  (ad+bc  ) h+  5c dl  ).yR  + (2abf  + (ad+bc  )„j  + 2cdm  )y„. 

+ (2abg+  (ad  + oc  )k+2cdn )y5] 

+y | [ (b  2e+bdh+d2l ) yr+ (b 2f +bd j +d 2m )y4+ (b  2g+bdk+d 2n )y  = ] = C . 
Introducing  the  condition  that  the  y xy  P_  term  shall  vanish  gives 

(8)  2a be  + (ad  + bc)h  + 2cdl  = 0 

2abf  + (ad  + bc)j  + 2cdm  = 0 

2abg  + (ad  + bc)k  + 2cdn  = 0 

But  from  the  conditions  for  orthogonality  cd=-ab  and  (8)  reduces 

to 

(9)  2ab(s-l)  + (ad+bc )h  = 0 
2ab(f— m)  + (ad  + bc  ).j  - 0 
2ab(g-n)  + (ad+bc )k  = 0 

From  (9)  follows  immediately 

(10)  _2ab_  _ _h_  = _i_  - _k_ 

1 — e 


ad+bc  1-e  m— f n-g 

Squaring  and  applying  a theorem  of  proportion  we  have 


. 


. 


28 


b.  inverse  of  Ruled  Cubic. 


(1) 


msider  a ruled  cubic  generated  by 
«X2+pA+y=0 
a - A b = 0 


where  a,?,v,a,b  are  linear  in  x,y,z.  The  equation  of  the  cubic 

i s 


(2) 


2 -i.  O 


cao  + vo 


It  involves  no  loss  of  generality  to  assume  a, ?, v homogeneous 

i n x , y , 2 . 

Now  apply  an  inversion 


(3 ) x : x 1 = y : y 1 = 


2 • v 1 — jj»  • y ! 2 y ! 2 7;  ! 2 — 2 .f.  y 2 -f.  g 2 * r* 


with  respect  to  a sphere  of  rsdius  r and  center  at  the  origin. 
Such  an  inversion  transforms  ~~y  plane  into  a so here  passing 
through  the  center  of  i nversi cr  and  in  oar  ici lar , if  the  plane 
passes  through  the  center  of  inversion  it  is  transformed  into 

itself » 

Since  «,  8,  y are  homogeneous  in  x,y,z  they  are  planes 

passing  through  the  center  of  inversion  and  are  therefore 

unchanged.  The  two  planes  a,b  are, however , transformed  into 

spheres  P and  Q,  Then  the  surface  into  which  the  cubic  is 

transformed  is  generated  by 

7.  a 2 + 8 a y 0 

D __  } — O 

where  P and  2 are  spheres  cassing  through  the  origin. 

But  this  ia  a quin tic  eyelids  with  the  center  of  the 
double  tangent  plans  on  the  nodal  circle. 

The  doable  line  of  the  ruled  cubic  which  is 
(5)  a = o = 0 


?Q 

-O  t/' 

is  transformed  into  the  intersection  of  P and  Q and  hence  is 
the  nodal  circle  of  the  quintic.  But  since  both  spheres  oass 
through  the  center  of  inversion  the  nodal  circle  passes  through 
that  point.  Since  through  the  original  center  of  inversion  on 
the  cubic  there  passes  a generator  of  the  cubic, then  there 
exists  a straight  line  on  the  quintic. 

Theorem  IS.  - - . enei  2 tin  circle  t is  s Dedal  quintic 
eyelids  passes  through  the  center  of  inversion  and  one  other 
point  on  the  nodal  circle. 

As  a further  particularization  the  center  of  inversion 
might  be  assumed  as  a poi  . the  nodal  line  of  the  ruled 
cubic.  In  that  case  t n s n o i 3. 1 line  as  a ? a i r a s t r s i g h t 1 i n e 
after  the  transformation  7 hi  is  the  planes,  of  the  cone  are 
transformed  into  spheres  and  the  new  surface  has  an  equation 

(6)  Pa2  + Qab  + Mb 2 - 0 
and  is  generated  by 

(7) 


PA2  + QA  + H = 0 


a - Ab  = 0 

But  this  is  a quartic  cyclide  with  a nodal  line. 

Inasmuch  as  the  auartic  cyciidss  have  been  rather  fully 
treated  in  the  literature  of  the  theory  of  surfaces  a further 
discussion  of  this  particular  surface  qculd  0 e out  of  place 

here  , 

Again, if  the  center  of  inversion  be  assumed  as  any  point 
in  space, the  cubic  will  be  transformed  into  a sextic  cyclide 

( Q 'S  p p 2 j.  P P P + p p 2 = 0 

V / *-  1-  p ' - 4 ^ ^ 


31 


II.  General  Cyclide. 

1.  Projective  Descriotion. 

Proceeding'  from  the  method  of  defining  the  auintic 
cyclide  we  shall  define. a general  cyclide  as  the  surface  generat- 
ed by  the  pencil  of  spheres 
( 1 ’i  p _ x o = r 

and  the  protectively  related  planes  of  the  developable 
(3)  cc0A?1  + o:1An'_1  + a?  Xn~~2  + ...  J>-  o:n  _ tA  + a?7  = C 

and  the  equation  of  the  general  cyclide  is  tree 

(3)  ff.0Pn  + c/5Pn-1C  - + ...  + <X„  — aPQ*-1  + = 0 

This  surface  has  the  sphere-circle  at  Infinity  and  the 
radical  circle  of  t h e p e n c i i.  c f s p h ares  as  a r. u 1 tide  c u r v s 
of  order  e .a  d t h ere  f c c e c e fc  c - s n e ; , t cl  a ss  c f s u r f aces 


si  oraer  t n + .;  r.a v i r ; - ; r 


y ■ c J 


' ' - 


Theorem.  The  developable  surface  (2)  is  doubly  tangent  to 
the  genera  l cyclide  (3). 

Any  two  planes  of  the  developable  surface  meet  in  a line 
which  intersects  the  surface  in  3n+l  points.  Each  plane  cuts 
the  cyclide  in  a circular  generator.  Put  as  the  planes  approach 
coincidence  the  two  circles  approach  coincidence  and  the  line 
approaches  a generator  of  the  developable.  Then  the  four  points 
in  which  the  line  cuts  the  two  circles  come  together  in  pairs 
so  that  the  generator  of  the  developable  surface  is  tangent 
to  the  cyclide  in  two  points. 

Again, a generator  of  the  developable  is  given  by  the 
equations 
(4) 


a o A ” + a a A ” ~ 1 + vPXn~2  + ... 
na0An_1  + (n-l)a1An~2  + ( n-2 ) a ? An~: 


' K n— a o 


, 32 

Eliminating  an  from  the  first  of  these  equations  and  the  equation 
of  the  eyelids  (3)  we  find 

(5)  (P-AQ)  [oi0Frl~1+(a0A  + o:1)P”~29+  . . . + (a0An“1+  . . , + an_5  )Qn~1]  =0 
Then  eliminating  an_4  from  the  second  equation  of  (4)  and  (5) 
we  have  f i n a 1 1 y 

) = 

where  i (a0,  aa,  . . . , h,  n,  A ) is  a function, homogeneous  of  degree 
n-2  in  P,Q  and  linear  in  a0, a1; . , . . This  equation  states  that 
the  generator  of  the  developable  surface  cuts  the 
two  double  points  which  lie  on  the  sphere  ?-A0=0  and  in  2n— 4 
points  which  lie  on  a eyelids  of  order  in—  , for  the  second 
factor  of  (Sf  is  of  the  same  form  as  (3)  but  of  degree  4 less. 


(6)  ( P-.X9  ) 2 f ( a o , a 3. , . . . , P,  Q , A ; = 


s t a t e s 

t h a t 

c y c 1 i c 

is  in 

a n d i n 

2 n— 4 

2.  Systems  cf  Curves  on  the  General  Cyclide. 

I!  he  ores  20.  guery  generatin'-*  plane  of  the  tangent  develop 
able  outs  the  general  cyclide  in  a circle  and  a rest  curve 
of  order  2n-l  which  lies  on  a cyclide  of  order  2n-l . 

The  proof  of  this  theorem  is  at  once  evident  from  the 

* 

examination  of  equation  (-5)  of  the  preceding  section, which 
can  be  written  in  the  form 

Cl)  CP—?  o ) f R C8  — 1 - 3 pn  — S'-.  + + a fan—  i,—  r- 

the  second  factor  of  which  is  of  the  same  form  as  i he  equation 
of  the  general  cyclide  (3)  of  the  last  section  but  lower  in 
degree  b y 2 . 

Theorem  21.  Any  so  here  cits  the  ■■■■ enerol  eye  line  in  o curve 
o f or  der  2n  + 2 -o n i c h c a n b e p r o :} e c l - o.  i ;■  .•  i o ■ • p i a n e c ,•  • v e o f 
o r a e r 2 n + 2 a n d ? e n u s n . 

The  equation  o f the  general  cyclide  may  be  written  in  the 

f o r m 

(2)  ao(^2-A)?I  + 0!1(R2-:5)»-i(a2-3)  + . . . + an  ( 32-3  )*  = 0 

ana  any  sphere 

(3)  R2  - 2 = : 

T hen  eliminati n g 1 2 o e t w sen  ( 2 } a n d ( 3 ) . . e b a v s 

t 0(C-A )»+a1(C-A)«”1( ;-F )+. . .+an_a ( - ( 0-3 )»“i+an ( C~B)n 

~ \J 

But  thisiis  the  equation  of  a ruled  (n+1 )-ic, generated  oy  the 
planes  of  c he  levs iopab le 

(5)  <x0A?I  + a-LXn~1  + ...  + an_1A  + a.n  = 0 
and  the  pencil  of  o lanes 

(6)  (G-A)  - A (0-6 ) = 0 


The  rulsd  surface  has  an  n-f old  11ns 
(7)  C-A  - 0-3  - 0 


34 


The  sobers  cuts  the  general  eyelids  in  the  curve  in  which  it 
cuts  the  ruled  (n+1)— ic.  'hen  the  curve  of  intersection  has 
2 n-f old  points.  If  this  curve  is  projected  on  a plane  from 
any  point  in  the  n-f old  line  of  the  ruled  surface, the  result 
is  a curve  of  order  2n+?  with  a 2n-fold  point  and  n ordinary 
doubly  points . (corresponding  to  the  n generators  of  the  ruled 
surface  passing  through  the  point  of  the  n-f old  line  used  as 
a center  of  projection). 

The  2 n-f old  point  counts  for 
in(|n-l)  = 2p2  _ n 

ordinary  double  points.  Then  this  with  the  n double  points 
makes  a count  of  3 n 2 double  points . but  the  maximum  number  of 
possible  double  points  is 

|2ni5-l2iin  + 2-2)  = ? n 2 n 


Then  the  deficieatcy  is 

2n 2 + n — 2n  2 — n , 

a result  which  we  have  already  seen  in  the  ouin tic.  eyelids 

where  n=2. 


By  a treatment  exactly  analagous  to  that  of  I, 2, a po  1C— 1 h 
there  is  developed  the 

Theorem  22.  The  general  eye  tide  can  be  Generated  in  a trivia 
infinite  number  of  ways  by  a recoil  of  (2n~l)-ics  and  a pencil 
of  concentric  spheres. 


J5 


3.  Locus  of 

the  Centers 

of  the 

Generating 

^ "I  V»|T» 

les . 

Theorem  2 

. The  locus 

of  the 

centers  of 

the 

general  in g 

circles  of  t h e 

general  eye 

lide  of 

order  2n  + l 

is  a 

space  curve 

of  order  2n+l. 

Consider  the  generating  plane 

(a)  «0  a71  + v.^71-1  + = o 

or 

(2)  xcp-t  + + zo,  + cp4  = 0 

where  the  <p 1 s have  the  same  s'*  gri^'ieanee  as  in  the  coin  tic 
eyelids  except  that  they  are  here  of  degree  n in  A.  The  corre- 
sponding sphere  of  the  pencil  is 

(3)  ( 1-A ) R 2 - ?'!f  tx  - ?p5y  - - 2»u  = 0 


where  the  1 s have  precisely  the  T.eanin- 


O U 1 n t,  1 


The  line  from  t hs 


inter  of  t he  sobers  perns ndicular  to  the 


corresponding  ilane  is 

( ■ ) Alz^izzTi  = ilz^llzl a - izz^lizi* 

?i  T ? <P3 


The  center  of  the  corresponding  circular  generator  is  the 
intersection  of  the  line  (45  with  the  plane  (2).  Solving  for 
the  coordinates  of  the  point  of  intersection  in  terms  of  A -we 

have 


( 5 ) c x 

py 

pz 

P t 


( 1-A  ) 33 1 cp 4 + © . , ( © 5 ;y  ? + cp n )-cp , ( cp  f + © | ) 

(1-A  )co?©4  + y?  + )“®j»  (Ti  + *I  ) 

(1-A  ) <P r, © 4 + <p o (©  i w i + ® 2 u/ ? )— cp,  (of + cp| ) 
(1-A ) (cpf  + cpi  + cpf ) 


But  these  equations  are  of  degree  2n+l  in  A and  hence  the  curve 
is  a space  curve  of  order  2n+l. 

The  locus  of  the  lines  perpendicular  to  the  generating 
planes  from  the  centers  of  the  corresponding  spheres  is  a ruled 


r»  o 

. V w 

of 

crdei 

? R + 

1. 

Co 

nsi 

d e r any 

plan 

a. 

(6) 

ax  + 

by 

+ c z 

a. 

d 

— o 

— ^ 

is 

no 

loss 

5 Of 

c?  0 n 

erali 

t y in 

choos 

ing 

c = 1 , T 

wee 

n e 

q uat ion 

(e) 

and  e 

q uat  i 

on  (4) 

we 

have 

(?) 

(1— A 

) a cp  P 

x + (1 

-a; 

)(© 

p.  + b cp  p 

)y+dcp? 

+ CO  p 1 

„ — o: „ v;  - 

c i— a ; 

) (a© 

l+?3 

)XH 

1 

— A ) b cr. 

+ a j 

A'  p - Cp  p © 3 

(7) 

t h 

ere  r s s u 

It'S 

(S) 

ox  = 

d©  a 

+ cp  J \<t 

!>■  - 

P ?.  ? 

i + b ( © 

t ©p-©? 

py  = 

d cp  P 

+ © P \n 

, r 

o r w 

p + a ( cc 

o VJ  x— <p  5 

\|/  2 ) 

ot  = 

-(1 

— A ) ( 

cp  3 - 

r 0 0 

1 + b cp  ? 

) 

ea 

uat 

ions 

are 

of 

dec 

Ire 

e n + 1 

i n A 

a nd 

hence 

fhess  equations  are  or  degree  n + j in  a ana  nence  t ns  surtace 
is  of  order  n+1.  The  sane  plane  (6)  intersects  the  oianes  of 
the  developable  surface  in  a class  curve  of  class  n » fence 
there  is  an  (( n, n+1 ] —correspondence  between  the  points  of  this 
curve  of  order  n+1  and  the  tangents  of  the  class  n-ic.  Tut 
in  such  a correspondence  there  are  n+(n+l)  coincidences  and 
hence  the  locus  of  the  centers  of  the  circular  generators 
.meets  this  olane  in  2n  + l ooints  and  is  therefore  of  order  2n+l. 


4 . Min 


Cyclide . 


ions ider  tfee  condition  that  a sphere  of  the  generating 


pencil  is  tangent 

to  t 

he  corresponding 

plane . 

1-A 

0 C 

V?/ 1 

o 

1-A  0 

1/? 

(1)  o 

0 1-A 

V.-9 

VI/ 1 

\’J  p VJ  r. 

\JJ  M 

T 4 

CD  2 Cp3 

This  is 

an  eouati 

on  of 

degree  2n  + 3 in  A 

since  the  cp 

1 s are  of 

degree  n 

and  the 

■■■11 ' 3 0 

f degree  1 . But  w 

hen  the  dete 

r m i n a n t is 

expanded 

it  turns 

cut 

that  1-A  is  a fac 

tor  sc  that 

A = j is 

one  root 

of  the  e 

0 U 3 t i 

on , 

Ther 

e for e one 

p a i r 

of  lines  is  r e a 1 

, name L . the 

line 

(2) 

(X  o ~ C!  j 
A - B = 

+ a ? 

Oi 

w 

+ . . . + an  = C 

and  the 

line  at  i n f i n i 

tv  in  the  plane 

(3) 

a o + a a 

+ a:  p 

+ . . . + an  = 0 

Corresponding  to  the  2n  + 2 other  solutions  of  equation  (1) 
there  are  2n+2  pairs  of  minimal  lines  on  the  general  cyclide. 

If  we  set  ud  the  condition  for  tangency  as  in  the  case  of 
the  auintic  cyclide  we  reach  the  condition 

(4)  [®  i¥a+<p?¥?  + ¥nW<,+  ( 1— A ) Cp  4 3 2=  [<pf  + tp|  + cp|3  [v/f  + u;§  + u;f + j 

and  this, we  see  at  once, is  of  degree  2n+2  in  A. 

I‘he  result  is  in  agreement  with  the  number  of  minimal  lines 
found  on  the  quintic  cyclide  in  which  n=2. 


. 

' 


3S7 

. Further  Generalized  Cyclides. 

Consider  the  surface  generated  b .y 

(1)  a0(AL,M,)7I  + a1(A,.u)n-1-i-...c(n  = 0 

where  A and  u are  connected  by  a rational  equation 

(2)  Rn(A,u)  = 0 
and 

(3)  F - AO.  = 0 

Equation  (l)  represents  a general  alg'eoraic  developable  surface. 
Each  plane  corresponding  to  a value  of  A cuts  the  corresponding 
sphere  of  the  pencil  (3)  in  a circle  and  the  totality  of  such 
circles  forms  the  system  of  circular  generators  of  the  eyelids. 

If  jj  — 0 we  have  the  particular  case  which  has  been  discussed  in 
the  preceding  pages. 

The  general  case  defined  by  equations  (1),(2)  and  (3) 
has  .yet  to  be  considered  in  detail. 

Again, consider  the  surface  generated  by 

(4)  P0An  + p,Ar‘-1  + PpA*-2  + ...  + Fn  = 0 

where 

( 5 ) P . = R 2 - A . 3 2 = x p + y 2 + z 2 ' . linear  in  x , y , z . 

and  the  pencil  of  spheres 
(e)  0,  - AQ?  = c- 

This  surface  is  a general  eyelids  of  order  2n+2  with  the  equation 


(7)  P0Q«  + 


-i-  P o 0 n~  i + p nn  = 

~ Yl i - i p - ? 


and  should  afford  a good  many  interesting  properties , and  would 
be  worth  invest i gat i n g . 

A still  further  generalization  would  be  to  let  the  o: 1 s of 
equation  (1)  represent  spheres  rather  than  planes. 


- 


i 


VITA 

Harvey  Pierson  Pettit, the  author  of  this  thesis, was  born 
April  25  1883  at  Cbadrcn, Nebraska . His  elementary  education 
was  received  in  the  public  schools  of  Michigan.  He  was 
graduated  from  Jackson  High  School, Jackson, Michigan  in  1810, 
and  from  Kalamazoo  College  in  1814  with  the  degree  of  A, 6, 
During  the  four  years, 1814-1818  he  was  teaching  mathematics 
in  the  high  school  at  Holland  Michigan.  He  was  Fellow  in  Math- 
ematics in  the  University  of  Kentucky  1816-18  receiving  the 
degree  of  M.A.  in  1818  from  the  University  of  Kentucky.  Ke 


was  an  assistant  and  graduate  student  in  Mathematics  at  the 
University  of  Illinois  1818-1822.  He  is  a member  of  the 
Mathematical  Association  of  America, Phi  Seta  Kappa,  and 


Xi 


u a t<  * 


t r : S ODDOF 


^ y 


i n g 


h a n K s 


tO  t hi 


members  cr  l r:  ? 


:•  3 C: as  worked,  and  ir. 


particular  to  Prof.  Arnold  Emch, under  whose  direction  this 
thesis  has  been  prepared, for  his  kindly  counsel  and  inspiration. 


